Problem: After leaving camp for a three-day journey into the frozen tundra, Daniela has just returned to camp. The three days of her journey can be described by displacement (distance and direction) vectors ${\vec{d_1}}$, ${\vec{d_2}}$, and ${\vec{d_3}}$, where each vector indicates Daniela's displacement from the start of her day to the end of her day. ${\vec{d_1}} = 8\hat{i} + 3\hat{j}$ ${\vec{d_2}}$ is not given. ${\vec{d_3}} = -3\hat{i} + 4\hat{j}$ (Distances are given in kilometers, $\text{km}$.) What distance did Daniela travel on day two?
Answer: Using our knowledge that Daniela starts at the camp on day one and returns to the camp at the end of day three, let's draw a better diagram of the situation. Now we can draw ${\vec{d_2}}$ so that connects Daniela's ending place on day one to her starting place on day three. Because Daniela returned to camp at the end of the third day, the sum of ${\vec{d_1}}$, ${\vec{d_2}}$, and ${\vec{d_3}}$ must be $(0,0)$. $\begin{aligned} {\vec{d_1}} + {\vec{d_2}} + {\vec{d_3}} &= (0,0)\\\\ {\vec{d_2}} &= -{\vec{d_1}} -{\vec{d_3}}\\\\ {\vec{d_2}} &= -(8\hat i + 3\hat j) - (-3\hat i + 4\hat j)\\\\ {\vec{d_2}} &= (-8\hat i - 3\hat j) + (3\hat i - 4\hat j)\\\\ {\vec{d_2}} &= -5\hat i + (-7)\hat j \end{aligned}$ We can find the magnitude of ${\vec{d_2}}$ (i.e., the distance Daniela traveled on day two) using the Pythagorean theorem. $\begin{aligned} \| {\vec{d_2}} \|^2 &= (-5)^2 + (-7)^2\\\\ \| {\vec{d_2}} \| &= \sqrt{25 + 49}\\\\ \| {\vec{d_2}} \| &= \sqrt{74}\\\\ \| {\vec{d_2}} \| &\approx 8.6 \text{ km} \end{aligned}$ ${\vec{d_2}}$ is pointing into the third quadrant, so we can find its direction (call it $\theta~$ ) using the arctangent function and adding $180^\circ$. $\begin{aligned} \tan \theta &= \dfrac{-7}{-5}\\\\ \theta &= \arctan{\left ( \dfrac{7}{5} \right )} \\\\ \theta &= 54^\circ \end{aligned}$ Adding $180^\circ$ to this result gives us the actual direction, $234^\circ$ (rounded to the nearest degree). Daniela traveled $8.6$ kilometers on day two. Daniela traveled in a direction of $234^\circ$ on day two.